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# Compressible Navier-Stokes Equations in Cylindrical Passages and General Dynamics of Surfaces—(I)-Flow Structures and (II)-Analyzing Biomembranes under Static and Dynamic Conditions

by Terry E. Moschandreou 1,*,† and Keith C. Afas 2,† 1
Department of Applied Mathematics, Faculty of Science, Western University, London, ON N6A 5C1, Canada
2
Department of Medical Biophysics, Faculty of Medical Science, Western University, London, ON N6A 5C1, Canada
*
Author to whom correspondence should be addressed.
These authors contributed equally to this work.
Mathematics 2019, 7(11), 1060; https://doi.org/10.3390/math7111060
Received: 31 August 2019 / Revised: 18 October 2019 / Accepted: 27 October 2019 / Published: 5 November 2019
A new approach to solve the compressible Navier-Stokes equations in cylindrical co-ordinates using Geometric Algebra is proposed. This work was recently initiated by corresponding author of this current work, and in contrast due to a now complete geometrical analysis, particularly, two dimensionless parameters are now introduced whose correct definition depends on the scaling invariance of the N-S equations and the one parameter $δ$ defines an equation in density which can be solved for in the tube, and a geometric Variational Calculus approach showing that the total energy of an existing wave vortex in the tube is made up of kinetic energy by vortex movement and internal energy produced by the friction against the wall of the tube. Density of a flowing gas or vapour varies along the length of the tube due to frictional losses along the tube implying that there is a pressure loss and a corresponding density decrease. After reducing the N-S equations to a single PDE, it is here proven that a Hunter-Saxton wave vortex exists along the wall of the tube due to a vorticity argument. The reduced problem shows finite-time blowup as the two parameters $δ$ and $α$ approach zero. A rearranged form for density is valid for $δ$ approaching infinity for the case of incompressible flow proving positive for the existence of smooth solutions to the cylindrical Navier-Stokes equations. Finally we propose a CMS (Calculus of Moving Surfaces)–invariant variational calculus to analyze general dynamic surfaces of Riemannian 2-Manifolds in $R 3$ . Establishing fluid structures in general compressible flows and analyzing membranes in such flows for example flows with dynamic membranes immersed in fluid (vapour or gas) with vorticity as, for example, in the lungs there can prove to be a strong connection between fluid and solid mechanics.
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Moschandreou, T.E.; Afas, K.C. Compressible Navier-Stokes Equations in Cylindrical Passages and General Dynamics of Surfaces—(I)-Flow Structures and (II)-Analyzing Biomembranes under Static and Dynamic Conditions. Mathematics 2019, 7, 1060.